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algebraic system (Definition)

Before defining what an algebraic system is, let's recall that, for a non-negative integer $ n$, an $ n$-ary operator $ f$ on a set $ X$ is a function of the form $ f:X^n\to X$. The number $ n$ is called the arity of $ f$. The most common $ n$-ary operators are those with arity $ \le 2$. An $ n$-ary operator is nullary (a constant) if $ n=0$, unary if $ n=1$, and binary if $ n=2$. Finally, a finitary operator on $ X$ is just an $ n$-ary operator on $ X$ for some non-negative integer $ n$.

Informally, an algebraic system consists of

a pair of sets $ (X,O)$ such that each element of $ O$ is a finitary operator on $ X$.

However, because we want to be able to compare various algebraic systems with the same $ O$, the dependency of $ O$ on $ X$ is undesirable. Therefore, to define an algebraic system formally, one employs the language of model theory, which requires two stages: (1) define what $ O$ is, and (2) define what $ (X,O)$ is.

  1. A set $ O$ is called an operator set if there is a function $ f$ from $ O$ to $ \mathbb{N}\cup \lbrace 0\rbrace$. Each element $ \omega\in O$ is called an operator symbol and $ f(\omega)$ is its arity.
  2. Given an operator set $ O$ and a set $ X$, the pair $ (X,O)$ is called an $ O$-algebra if there is a set $ O_X$ such that
    • $ O_X$ consists of finitary operators on $ X$
    • there is a one-to-one correspondence between $ O$ and $ O_X$, with mapping $ \omega\mapsto \omega_X$, such that the arity of $ \omega_X$ is the arity of the operator symbol $ \omega$
    An algebraic system (variously called an algebraic structure, universal algebra, or simply algebra), is an $ O$-algebra $ (X,O)$ for some set $ X$ and some operator set $ O$.

In an algebraic system $ (X,O)$, $ X$ is called the underlying set of $ (X,O)$, and $ O_X$ the operator set of $ (X,O)$. When there is no confusion, we shall simply say that $ \omega$ (instead of $ \omega_X$) is an operator on $ X$. If the underlying set $ X$ is a singleton, then $ (X,O)$ is called a trivial algebraic system.

Remarks

  1. In any algebra $ X$, we typically adopt the convention that the constant operators are identifed with their values in $ X$. Also, we write $ x^*$ as the value of a unary operator $ *$ applied to $ x\in X$. Finally, $ x\circ y$ denotes the value of a binary operator $ \circ$ applied to $ (x,y)\in X^2$.
  2. Two algebraic systems $ (A,O_1)$ and $ (B,O_2)$ are of the same type, or signature, if $ O_1=O_2$. A set of algebraic systems is of the same type if any two of them are of the same type.

Examples.

  1. If $ O=\varnothing$, then an $ O$-algebra is any set, empty or not. If $ O$ is a singleton whose sole element has arity 0, then an $ O$-algebra is a non-empty set, and in fact a pointed set.
  2. A group $ G$ is an algebraic system with three operators: a constant operator $ e$, a unary operator $ -1$, and a binary operator $ \cdot$ called the multiplication. Of course, these operators satisfy additional conditions.
  3. A ring $ R$ without a multiplicative identity is an algebraic system with one constant operator 0, a unary operator $ -$, and two binary operators $ +$ and $ \cdot$. As in the previous examples, these operators meet certain additional constraints before $ R$ can be called a ring. For example, $ R$ with $ O=\lbrace 0, -, +\rbrace$ form a group (in fact, Abelian group).
  4. A ring $ R$ with a multiplicative identity is an algebraic system. Like the previous examples, it has operators $ 0,-,+,\cdot$. In addition, it has a second constant operator $ 1$, called the multiplicative identity.
  5. A field $ F$ is not an algebraic system. The multiplicative inverse $ a^{-1}$ of an element $ a$ is not defined when $ a=0$. It is not an operator, only a partial operator. It is possible to artificially define the inverse of 0 to be 0, which makes a field into an algebraic structure. However, the drawback is that it violates the rule that $ aa^{-1}=1$ (which means we are no longer looking at a field!) A field is an example of a partial algebraic system.
  6. A lattice $ L$ is an algebraic system with two binary operators $ \vee$ and $ \wedge$. A bounded lattice is also an algebraic system. In addition of being a lattice, it has two constant operators 0 and $ 1$.
  7. A complete lattice is not an algebraic system, since arbitrary meet and join are infinitary operators, that is, the domain has the form $ X^I$, where $ I$ is some arbitrary set of cardinality not necessarily finite.
  8. So far, all of the algebraic systems listed above have finite operator sets. Here's an example where the operator set $ O$ is possibly infinite. Let $ R$ be a ring and $ M$ a (left) module over $ R$. Then $ M$ is an abelian group (with operators $ 0,\,+$). In addition, for each $ r\in R$, we can define a unary operator $ \cdot_r$ on $ M$ by left multiplication $ \cdot_r(m)=r\cdot m$. So, $ M$, together with $ O=\lbrace 0,+,\cdot_r\mid r\in R\rbrace$ constitute an algebraic system.

Remark. All of the examples are trivially algebraic structures, if we “forget” one or more (or all) of the operators. For example, a field is by definition a ring with an additional operator (multiplicative inverse). Therefore, as a ring, a field is an algebraic structure. But as a field, it is not. Formally, if $ O^{\prime}\subseteq O$, then any $ O$-algebra is an $ O^{\prime}$-algebra.

Bibliography

1
А. И. Мальцев: Алгебраические системы. Издательство ``Наука''. Москва(1970).
2
P. M. Cohn: Universal Algebra, Harper & Row, (1965).
3
G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
4
P. Jipsen: Mathematical Structures: Homepage



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"algebraic system" is owned by CWoo. [ full author list (4) | owner history (1) ]
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See Also: relational system

Other names:  algebraic structure, universal algebra, signature, trivial algebra
Also defines:  $n$-ary operator, finitary operator, infinitary operator, operator set, constant operator, operator symbol, nullary operator, type, trivial algebraic system

Attachments:
homomorphism between algebraic systems (Definition) by CWoo
subalgebra of an algebraic system (Definition) by CWoo
direct product of algebras (Definition) by CWoo
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Cross-references: module, infinite, finite, cardinality, domain, join, arbitrary meet, complete lattice, bounded lattice, lattice, partial algebraic system, inverse, partial operator, multiplicative inverse, field, abelian group, meet, multiplicative identity, ring, satisfy, group, pointed set, singleton, algebra, mapping, one-to-one correspondence, model theory, binary, unary, arity, number, function, operator, integer
There are 89 references to this entry.

This is version 34 of algebraic system, born on 2006-03-07, modified 2007-08-31.
Object id is 7695, canonical name is AlgebraicSystem.
Accessed 9207 times total.

Classification:
AMS MSC: 08A62 (General algebraic systems :: Algebraic structures :: Finitary algebras)
 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
 08A05 (General algebraic systems :: Algebraic structures :: Structure theory)
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